Pre-coding and decoding polar codes using local feedback

ABSTRACT

Disclosed are devices, systems and methods for precoding and decoding polar codes using local feedback are described. One example method for improving an error correction capability of a decoder includes receiving a noisy codeword vector of length n, the codeword having been generated based on a concatenation of a convolutional encoding operation and a polar encoding operation and provided to a communication channel prior to reception by the decoder, performing a successive-cancellation decoding operation on the noisy codeword vector to generate a plurality of polar decoded symbols (n), generating a plurality of information symbols (k) by performing a convolutional decoding operation on the plurality of polar decoded symbols, wherein k/n is a rate of the concatenation of the convolutional encoding operation and the polar encoding operation, and performing a bidirectional communication between the successive-cancellation decoding operation and the convolutional decoding operation.

CROSS-REFERENCE TO RELATED APPLICATION

This patent document claims priority to and benefits of U.S. ProvisionalPatent Application No. 62/775,266 entitled “PRE-CODING AND DECODINGPOLAR CODES USING LOCAL FEEDBACK” and filed on 4 Dec. 2018. The entirecontent of this patent application is incorporated by reference as partof the disclosure of this patent document.

TECHNICAL FIELD

This document generally relates to error correction codes, and moreparticularly to pre-coding and decoding polar codes using localfeedback.

BACKGROUND

A communications system generally adopts channel encoding to improvereliability of data transmission and ensure quality of communications inthe presence of various types of noise and errors. Polar coding is ageneral and extremely powerful error-correction technology proposedaround a decade ago, and which is currently used for coding the controlchannels in the eMBB mode of the Fifth Generation (5G) wirelessstandard. In addition to wireless communications, polar codes may haveapplications in fiber-optic networks, data storage, satellitecommunications, and more.

SUMMARY

Embodiments of the disclosed technology relate to methods, devices andsystems for improving an error correction capability of an encoder anddecoder based on pre-coding and decoding polar codes using localfeedback. The methods and devices described in the present documentadvantageously, among other features and benefits, reduce acommunication receiver's complexity while realizing the potential of theperformance of polar codes.

In an example aspect, a method for improving an error correctioncapability of a decoder includes receiving a noisy codeword vector oflength n, the codeword having been generated based on a concatenation ofa convolutional encoding operation and a polar encoding operation andprovided to a communication channel or a storage channel prior toreception by the decoder, wherein n is a positive integer, performing asuccessive-cancellation decoding operation on the noisy codeword vectorto generate a plurality of polar decoded symbols that comprises aplurality of convolutionally encoded symbols (n₁), a first plurality ofinformation symbols (k−n₁), and a plurality of frozen symbols (n−k),wherein k and n₁ are non-negative integers, generating a secondplurality of information symbols (k₁) by performing a convolutionaldecoding operation on the plurality of convolutionally encoded symbols,wherein k₁ is a non-negative integer, wherein k₁/n₁ is a rate of theconvolutional encoding operation, and wherein (k₁+k−n₁)/n is a rate ofthe concatenation of the convolutional encoding operation and the polarencoding operation, and performing a bidirectional communication betweenthe convolutional decoding operation and the successive-cancellationdecoding operation, wherein the bidirectional communication comprisesdecoding information.

In another example aspect, a method for improving an error correctioncapability of an encoder includes receiving a plurality of informationsymbols (k), wherein k is a positive integer, generating a plurality ofconvolutionally encoded symbols (n) by performing a convolutionalencoding operation on the plurality of information symbols and aplurality of frozen symbols (n−k), wherein n is a positive integer,generating a plurality of polar encoded symbols by performing a polarencoding operation on the plurality of convolutionally encoded symbols,wherein the polar encoding operation is based on a transform, andwherein a rate of the polar encoding operation is one, and providing theplurality of polar encoded symbols for transmission or storage.

In yet another example aspect, a method for improving an errorcorrection capability of an encoder includes receiving a noisy codewordvector of length n, the codeword having been generated based on aconcatenation of a convolutional encoding operation and a polar encodingoperation and provided to a communication channel or a storage channelprior to reception by the decoder, wherein n is a positive integer,performing a successive-cancellation decoding operation on the noisycodeword vector to generate a plurality of polar decoded symbols (n),generating a plurality of information symbols (k) by performing aconvolutional decoding operation on the plurality of polar decodedsymbols, wherein k is a positive integer, and wherein k/n is a rate ofthe concatenation of the convolutional encoding operation and the polarencoding operation, and performing a bidirectional communication betweenthe successive-cancellation decoding operation and the convolutionaldecoding operation.

In yet another example aspect, the above-described methods may beimplemented by an apparatus or device that comprises a processor and/ormemory.

In yet another example aspect, these methods may be embodied in the formof processor-executable instructions and stored on a computer-readableprogram medium.

The subject matter described in this patent document can be implementedin specific ways that provide one or more of the following features.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating an example embodiment of anencoder structure for a polar code.

FIG. 2 is a block diagram illustrating another example embodiment of anencoder structure for a polar code.

FIG. 3 illustrates the frame error rate (FER) of a polar codeconcatenated with a number of convolutional codes with differentlengths.

FIG. 4 illustrates a high-level description of an SC decoding algorithm.

FIG. 5 illustrates a high-level description of another SC decodingalgorithm.

FIG. 6 illustrates the FER of polar codes with a genie-aided SC decodingalgorithm.

FIG. 7 is a block diagram illustrating an example of a Viterbi-aidedsuccessive cancellation (SC) decoder structure for a polar code.

FIG. 8 illustrates an example snapshot of the Viterbi algorithm.

FIG. 9 illustrates an example of feedback instructions for resetting theSC decoder.

FIG. 10 illustrates an example of instructions for the SC decoder.

FIG. 11 illustrates the FER of polar codes with different SC decodingalgorithms.

FIG. 12 illustrates a flowchart of an example method for encoding polarcodes.

FIG. 13 illustrates a flowchart of an example method for decoding polarcodes.

FIG. 14 illustrates a flowchart of another example method for decodingpolar codes.

FIG. 15 is a block diagram representation of a portion of an apparatus,which can be used to implement some embodiments of the presentlydisclosed technology.

DETAILED DESCRIPTION

Polar codes are a new approach to maximizing the rate and reliability ofdata transmissions, and have been adopted to improve coding performancefor control channels in 5G. At the same time, they reduce the complexityof design and ensure service quality. Polar codes are a type of linearblock error correcting code, whose code construction is based on amultiple recursive concatenation of a short kernel code which transformsthe physical channel into virtual outer channels. When the number ofrecursive concatenations becomes large, the virtual channels tend toeither have very high reliability or very low reliability (in otherwords, they polarize), and the data bits are allocated to the mostreliable channels.

Embodiments of the disclosed technology relate to precoding and decodingpolar codes using local feedback, thereby reducing a communicationreceiver's complexity while realizing the potential of the performanceof polar codes. The disclosed embodiments can, for example, be used inany communication or data-storage system that is affected by noise anduses polar coding to correct errors. The disclosed embodiments may beparticularly attractive in systems that already use polar coding, butcan ill-afford higher complexity decoding algorithms such as CRC-aidedlist decoding. From a different viewpoint, the present document includesmethods for improving performance of successive-cancellation decodingwithout significantly increasing the overall computationalcomplexity/requirements. The value of polar codes is inherently boostedfor many scenarios based on this improved and additional functionality.

Section headings are used in the present document to improve readabilityof the description and do not in any way limit the discussion orembodiments (and/or implementations) to the respective sections only.

Introduction to Polar Codes and Successive Cancellation Decoding

Polar codes. In the present document, embodiments of the disclosedtechnology use a (n, k) polar encoder, where n=2^(m) corresponds to mlevels of polarization and k denotes the number of information bits inthe polar code, resulting in n−k frozen bits. In some embodiments, thefrozen bits are set to zero. In other embodiments, the frozen bits arecomputed such that they are known a priori at both the transmitter andthe receiver. The relationship between the transmitted symbols {x_(i)}and the uncoded information bits {u_(i)} is given by:

${x = {uG}_{n}},{G_{n} = {{B_{n}\begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}}^{\otimes m}.}}$

Herein, x=uG_(n) is referred to as a polar transformation, and G_(n) isthe polar code generator (or generating) matrix. In some embodiments,the polar code generator matrix is the m-th Kronecker power of the 2×2kernel matrix that is multiplied by a length-n bit reversal matrix B_(n)and u denotes the uncoded information bit vector that includes kinformation bits and n−k frozen bits. In an example, the 2×2 kernelmatrix (denoted F) and the m-th Kronecker power of the kernel matrix form=3 are given by:

${F = \begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}},{{{and}\mspace{14mu} F^{\otimes 3}} = {\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\end{bmatrix}.}}$

In some embodiments, different 2×2 kernel matrices

$\left( {{e.g.},\begin{bmatrix}1 & 1 \\1 & 0\end{bmatrix}} \right)$may be used to generate the polar code. In other embodiments, the bitreversal matrix B_(n) may be omitted in the generation of the polarcode.

Successive cancellation (SC) decoding. SC decoding provably enablespolar codes to achieve the capacity of memory symmetric channels. In anexample, successive cancellation decoding starts by using the symbolsreceived over the channel to decode a first bit, and then subsequentbits are decoded based on the symbols received over the channel and oneor more previously decoded bits. For example, a tenth bit is decoded asa function of the symbols received over the channel and the valuesdetermined (e.g., “0” or “1”) for at least one of the nine previous bitsthat have already been decoded.

In some embodiments, successive cancellation decoding comprises decodingu_(i) for i=1,2, . . . , n sequentially while assuming the values of theprevious u_(i)'s. In other words, to decode u_(i), it is assumed thatthe previous bits u₁, . . . , u_(i−1), are all known (or correctlydecoded) and hence available to the decoder similar to the channelobservation vector y.

That is, given the channel observation vector y, the successivecancellation decoder estimates û₀, û₁, . . . , û_(n−1) one-by-one byfirst efficiently calculating the following pair of probabilities ateach step:P(û _(i)−0|y,u ₀ ^(i−1)) andP(û _(i)=1|y,u ₀ ^(i−1)).

Then, a decision is made on û_(i) ∈{0,1}.

In some embodiments, successive cancellation decoding can leverage thebutterfly structure of polar codes to efficiently calculate theprobability pair shown above for i=0,1, . . . , n−1. Upon calculation ofthe probability pairs, the decision on û_(i) is prioritized by firstlooking up the available frozen values, and then the freshly calculatedprobabilities.

Examples of Encoder Structures for Embodiments of the DisclosedTechnology

Embodiments of the disclosed technology provide concatenations of polarcodes and convolutional codes, which advantageously enable the Viterbialgorithm to be used in the successive cancellation decoding of polarcodes.

In some embodiments, let n denote the length of the uncoded vector u,which consists of n−k frozen indices (denoted

) and k indices (denoted

) that are typically used for information bits (or symbols). Theembodiments described herein use a convolutional code to precode k₁information bits into n₁=k coded ones. These k bits are placed in theindices in the set

, and sent to a polar encoder along with the remaining information bits.This advantageously results in a concatenation between an [n₁, k₁]convolutional code and a [n, k=n₁] polar code.

FIG. 1 is a block diagram illustrating an example embodiment of anencoder structure for a polar code. As illustrated therein, a terminatedconvolutional code of length n₁ is generated from a sequence of k₁information bits. The uncoded vector u (with regard to the polar code)is generated by multiplexing these k=n₁ bits with the n−k frozen bits.The multiplexed length-n vector u is then polar coded to form thelength-n polar codeword x. The rate of the concatenated polar codeillustrated in FIG. 1 is k₁/n.

In some embodiments, the frozen bits may be fixed to all zeros or allones. In other embodiments, the frozen bits may be set to apredetermined patterns of zeros and ones. In yet other embodiments, thefrozen bits may be dynamically determined based on values of other bitsthat are multiplexed before that specific frozen bit.

In an example, the convolution code in FIG. 1 may be a rate-1/2convolutional code, in which case the rate of the concatenated polarcode illustrated in FIG. 1 is k₁/n˜k/2n.

In an example, the convolutional code may have a generator matrix givenbyG _(conv)=[1+D ²,1+D+D ²].

Convolutional codes provide a natural local error correction capability,which can be utilized as a genie-like aid provided for successivecancellation decoder. The traceback depth in convolutional codesdetermines the required delay to validate a bit from the receivedsequence by Viterbi Algorithm. It is known that the successivecancellation of polar codes suffers from the error propagationphenomena, i.e., when it makes the first mistake during the decodingprocess, it is bound to make a large number of additional mistakes onthe average. This property translates to a poor bit error rate (BER) forpolar codes in general. Hence, it is desired to utilize a convolutionalcode with low traceback depth in order to increase the chance ofcorrecting an error before future incorrect bits appear.

The traceback depth is estimated to have a linear relation with theconstraint length of convolutional codes. However, precoding withconvolutional codes with large constraint lengths has its own merits:

The free distance, denoted by d_(free), measures the error correctioncapability of convolutional codes; and, the common constructions ofconvolutional codes with large d_(free) usually require a largeconstraint length.

A large rate loss cannot be tolerated only to add a second layerprotection for the information bits, since a simple reduced-rate polarcode may show a better performance in fair comparisons. On the otherhand, common constructions of convolutional codes with high rates alsorequire larger constraint lengths.

In some embodiments, not all of the k bits in

require the extra protection of the convolutional code. In fact, most ofthe indices in

correspond to almost-noiseless bit channels. Accordingly, one can modifythe encoder structure in FIG. 1 to protect only a small portion of theinformation bits with convolutional codes. This modificationadvantageously enables the use of convolutional codes with low tracebackdepths and simulates the polar genie more efficiently. FIG. 2illustrated the improved encoder structure.

Embodiments of the disclosed technology include selecting an optimalvalue for n₁ to provide a second layer of error protection for thenoisiest bits. In some embodiments, the optimal value can be numericallydetermined. FIG. 3 illustrates the frame error rate (FER) of an [8192,4096] polar code concatenated with a number of convolutional codes withdifferent lengths. In FIG. 3, k₁ denotes the length of the informationbit sequence fed to the convolutional code, and the optimal codelengthin this example is given by n₁ ^(opt)=108.

In some embodiments, the selection of the number of the frozen values(nominally n−k) and their multiplexing (as illustrated in the encoderstructures in FIGS. 1 and 2) is similar to rate selection. That is, therate of the concatenation of the convolutional code and the polar codecan be adjusted by varying the number of frozen values selected andmultiplexed into the input of the polar code.

In some embodiments, the location of the frozen bits (or symbols) can beselected based on the noise level of the corresponding bit-channel,which itself depends on the noise level of the communication channel. Inan example, the n−k noisiest bit-channels are frozen, leaving the k lessnoisy ones for the information bits. Multiple algorithms may be used totrack these noise levels very efficiently even for large values of n.Some algorithms that can be used include constructions based on Gaussianapproximation and channel degradation.

In some embodiments, as discussed above, the polar encoding may beimplemented using a generator matrix or a recursive butterfly structure.In an example, the polar encoding may use only a portion of rows of thegenerator matrix (G). In some embodiments, the rows of G that are usedin the polar encoding process are selected based on their correspondingBhattacharya parameters or Hamming weight of the rows of the matrix.

Examples of decoding structures for embodiments of the disclosedtechnology

As described above, successive cancellation decoding is commonly used todecode polar codes, and a high-level description of the SC decodingalgorithm is illustrated in FIG. 4. As illustrated therein, the input isa received vector y and the output is the decoded vector û. Thealgorithm includes first computing the probabilities P(û_(i)=0|y,u₀^(i−1)) and P(û_(i)=1|y,u₀ ^(i−1)) for each bit (or symbol) of thereceived vector. As discussed above, if u_(i) is frozen, then û_(i) isset to the value of frozen bit (since it is known at both the encoderand the decoder). Otherwise, the value of û_(i) is based on comparingthe values of the probabilities.

The embodiments of successive-cancellation decoding described herein aredesigned to correct the received bits/symbols in a sequential fashion.Due to lack of feedback, traditional successive-cancellation decodingsuffers significantly from error-propagation. In particular, at thereceiver, there is no mechanism to correct an erroneous decision on adecoded bit/symbol once it happens. This single erroneous decision thenadversely affects many future decisions in the successive-cancellationdecoder. As described herein, embodiments incorporate an internalfeedback mechanism into a polar decoding algorithm.

Additional details regarding successive cancellation decoding of polarcodes may be found in U.S. Pat. No. 9,176,927, which is herebyincorporated by reference, in its entirety, as part of this application.

In some embodiments, polar codes may be decoded using a list decodingalgorithm, which provides a list of L highly likely candidates for û.Existing implementations of list decoding show that selecting the mostlikely candidate from the list brings the error rate down to nearoptimal value (ML) even when small values of L are taken. However, byslightly modifying the structures polar codes by precoding the kinformation bits with a cyclic redundancy check (CRC), unverifiedcandidates from the list can be first rejected and then the ML selectionmade. Hence, the CRC acts like a genie that informs the decoder aboutinvalid codewords.

As discussed above, the decoding performance of polar codes is improvedby leveraging the frozen values (in a genie-like manner) and may befurther improved by using the genie (or a CRC-aided implementation) tohelp with the decoding of the non-frozen bits by correcting the wronglydecoded ones for a limited number of times.

However, the CRC-aided list decoding of polar codes (e.g., as describedin U.S. Pat. No. 9,503,126) is characterized by the complexity ofdecoding growing proportionally to the list size. Thus, CRC-aided listdecoding may be impractical if the required list size is too largeand/or if the computational resources of the system are severelylimited. In contrast, the complexity of decoding in the disclosedembodiments remains almost unchanged as compared to conventionalsuccessive-cancellation decoding. In fact, this complexity depends onlyon the number of decision errors expected in the worst case.

Examples of Genie-Aided SC Decoding of Polar Codes

In some embodiments, the genie-aided SC decoding of polar codes assumesthat the SC decoder is equipped with some side information that helpscorrecting its first mistake (if there is any) during the decodingprocess. The probability of successful decoding is then given by

$P_{success} = {\underset{\underset{{no}\mspace{14mu}{genie}\mspace{14mu}{needed}}{︸}}{\prod\limits_{i \in \mathcal{J}}\left( {1 - p_{i}} \right)} + {\underset{\underset{1\text{-}{genie}\mspace{14mu}{needed}}{︸}}{\sum\limits_{i \in \mathcal{J}}{p_{i}{\prod\limits_{{j \in \mathcal{J}},{j \neq i}}\left( {1 - p_{j}} \right)}}}.}}$

The FER in this case (e.g., correcting up to 1 mistake) is thenexpressed as

$P_{e}^{(1)} = {1 - {\left( {\prod\limits_{i \in \mathcal{J}}\left( {1 - p_{i}} \right)} \right){\left( {1 + {\sum\limits_{i \in \mathcal{J}}\frac{p_{i}}{1 - p_{i}}}} \right).}}}$

Similarly, the genie may be used to correct γ errors. A high-leveldescription of SC decoding algorithm that uses the genie up to γ timesis illustrated in FIG. 5. The algorithm in FIG. 5 operates similarly tothe algorithm illustrated in FIG. 4, but additionally includes settingthe value of û_(i) for a non-frozen bit based on the correctioncapability of the genie.

FIG. 6 illustrates the FER of polar codes with a genie-aided SC decodingalgorithm that is used to correct an increasing number of errors(plotted for γ=1, . . . , 4). As illustrated therein (for a[1024,512]-polar code), even for a very limited number of mistakescorrected, the performance of polar code decoding improves drastically.

Examples of Viterbi-Aided SC Decoding of Polar Codes

In some embodiments, the Viterbi-aided SC decoding of polar codesassumes that

∪

={0,1, . . . , n−1} in which

denotes the location of the unfrozen indices in u. It is further assumedthat

_(conv)={σ₀, . . . , σ_(n) ₁ ⁻¹}⊂

corresponds to the indices in which the length-n₁ convolutional code islocated, and that σ₀<σ₁< . . . <σ_(n) ₁ ⁻¹. The Viterbi-aided SC decoderis an addition on top of the conventional SC decoder that reduces theoverall error rate by decreasing rate of the decoding mistakes made onthe bit-channels in

_(conv).

In an example, and given the received vector y from the channel, theViterbi-aided SC decoding of polar codes starts by estimating û₀, û₁, .. . one-by-one. After estimating û_(i), two different cases may appear:

i ∈(

∪

)\

_(conv): decoder continues the process normally as it would have done inthe absence of the convolutional code.

i ∈

_(conv) or equivalently i=σ_(j) for some j: the freshly estimated valueof û_(σ) _(j) is fed to the Viterbi decoder for second step validations.

If Viterbi decoder discovers any disparities on û_(σ) _(j) , or thoseprovided earlier from the SC decoder, i.e. û_(σ) _(j) , l≤j, a feedbackarm gets activated. Upon activation of the feedback, the correct valueof û_(σ) _(j) , denoted by û_(σ) _(j) , is sent back to the SC decodingblock. The SC decoder then resets its index back to σ_(l), and restartsthe calculations from there by considering the invalidated values forû_(σ) _(l) that are provided by the Viterbi decoder. In someembodiments, the Viterbi decoder is configured to keep and updated alist of invalidated symbols that it shares with thesuccessive-cancellation decoder.

An overview of the Viterbi-aided SC decoding algorithm is provided inFIGS. 7-10.

As illustrated in FIG. 7, the output from SC decoder is split into twosequences: unprotected less-noisy bits, and n₁ bits that form theconvolutional codeword for the second step verification. In someembodiments, the decoder structure in FIG. 7 is a joint convolutionaland polar decoder, which combines convolutional decoding and polardecoding. As described herein, the convolutional decoder is configuredto robustly identify errors that have occurred either in the previousblock or in previous symbols of the current block. That is,convolutional decoding provides error detection, which is advantageouslyused to restart the SC decoder in order to improve overall decodingperformance.

In some embodiments, the feedback mechanism only gets activated if thesecondary error-detection module catches an inconsistency in the outputof the SC decoder. When the signal-to-noise ratio is high, the SCdecoder often makes no decision errors at all. Hence, the averagedecoding complexity remains unchanged. In addition, an appropriatechoice of the secondary error-detection module (e.g., decoding over aconvolutional trellis or tree that is associated with the convolutionalencoder on the transmitter side, the Viterbi algorithm, theforward-backward algorithm (FBA), or the BCJR algorithm) makes itpossible to perform the verification step very efficiently.

FIG. 8 illustrates an example snapshot of the Viterbi algorithm, whereinthe sample trellis depicts the delay in symbol verification. A symbol isverified when all of the trellis paths agree on it. The feedback isactivated when a symbol is verified to be incorrectly estimated bysuccessive cancellation.

The example illustrated in FIG. 8 assumes that the input symbol for theViterbi algorithm is formed of 2 bits. The input and output sequences ofthe Viterbi decoder are denoted as û_(σ) ₀ , û_(σ) ₁ , û_(σ) ₂ , û_(σ) ₃, . . . and {circumflex over (û)}_(σ) ₀ , {circumflex over (û)}_(σ) ₁ ,{circumflex over (û)}_(σ) ₂ , {circumflex over (û)}_(σ) ₃ , . . . ,respectively. As illustrated in FIG. 8, there is some delay between thelast received input symbol and the most recent verified one, which isstatistically bounded by traceback depth of the convolutional code.

When a mismatch between input and output symbols is discovered, i.e.,{circumflex over (û)}_(σ) _(l−1) {circumflex over (û)}_(σ) _(l) ≠û_(σ)_(l−1) û_(σ) _(l) , the feedback mechanism adds the incorrect symbolû_(σ) _(l−1) û_(σ) _(l) to the list of blocked symbols for indices{σ_(l−1), σ_(l)}, and the SC decoding process is restarted from indexσ_(l−1) (as illustrated in FIG. 9).

In some embodiments, the immediate replacement of û_(σ) _(l−1) û_(σ)_(l) with û_(σ) _(l−1) û_(σ) _(l) is not allowed when a mismatch occurssince the latter is a function of the input sequence and has to bere-calculated according to the new input symbols. Instead, the SCdecoder restarts the decoding process at σ_(l−1) by selecting the nextmost likely unblocked symbol. To determine the next most likely symbol,we value of the less reliable bit-channel is first flipped. If thatcomes back unverified as well, the algorithm proceeds by flipping thevalue of the more reliable bit-channel; and, if they both returnunverified, the SC decoder flips both of them. The various operatingcases are enumerated in FIG. 10.

In an example, it is assumed that the SC decoder is reset to indexσ_(l−1) and is provided with an ordered set of blocked symbols {αβ, ˜αδ}from the Viterbi decoder, where ˜α denotes the flipped value of a.Further assume that α_(l−1) corresponds to the less reliable bit-channelbetween the two. It is observed that both values of α, ˜α got rejectedfrom the Viterbi decoder. Hence, the chances are that the decodingmistake was made on the more reliable bit-channel in the first place. SCdecoder then proceeds by keeping û_(σ) _(l−1) =α and flipping thedecision on the next bit, i.e. û_(σ) _(l) =˜β.

In some embodiments, the Viterbi algorithm also accepts soft information(symbol likelihoods) as input. In other embodiments, the SC decoder isalso capable of calculating the likelihoods for the bits in its outputsequence. In yet other embodiments, the current decoding scheme can beimproved by feeding the calculated soft information from the SC decoderto the Viterbi decoder instead of the hard decisions.

Example Simulation Results

FIG. 11 illustrates the comparison between the Viterbi-aided SC decodingalgorithm and a conventional [8192, 4096]-polar code under SC decoding.A noticeable improvement is observed particularity at high SNR regime.The lower bound (LB) curve corresponds to a genie-aided SC decoder,which is enabled on all those bit-channels that belong to theconvolutional codeword. In other words, it prevents SC decoder frommaking any decoding mistakes on those n₁ bit-channels in

_(conv).

Decoding Complexity Analysis

The decoding complexity of the Viterbi algorithm is an asymptotic linearfunction of n₁. Furthermore, the Viterbi decoding block never activatesthe feedback if the message is correctly estimated by successivecancellation itself. The performance of the SC decoder in FIG. 11indicates that this event, for instance, happens with very highprobability [Pr.>0.99] at an SNR of 2 dB. Furthermore, a singleiteration fixes the error in most of the cases, in which successivecancellation made a mistake.

Some embodiments of the disclosed technology can be implemented asconfigurations of an error correction decoder that allow the output of aconventional polar successive-cancellation decoder to be fed into asecondary error-correction decoding module on the fly. This secondarymodule provides feedback to the successive-cancellation decoder duringthe decoding process indicating which decisions on individualbits/symbols may be in error.

Feedback mechanism of the example embodiments described in the presentdocument can be applied to combine successive-cancellation decoding ofpolar codes with a secondary error-detection module that allows decodingthe received bits/symbols locally on the fly. The secondaryerror-detection module can be implemented using any one of a pluralityof error-detection techniques or algorithms, such as a Viterbi decodingalgorithm that is used for decoding convolutional codes.

Embodiments of the disclosed technology can be applied in anycommunication or data-storage system that is affected by noise and usespolar coding to correct errors. The disclosed embodiments may beparticularly attractive in systems that already use polar coding, butcan ill-afford higher complexity decoding algorithms such as CRC-aidedlist decoding.

FIG. 12 illustrates a flowchart of an example method 1200 for polarencoding. The method 1200 includes, at operation 1210, receiving a noisycodeword vector of length n, the codeword having been generated based ona concatenation of a convolutional encoding operation and a polarencoding operation and provided to a communication channel or a storagechannel prior to reception by the decoder, wherein n is a positiveinteger.

The method 1200 includes, at operation 1220, performing asuccessive-cancellation decoding operation on the noisy codeword vectorto generate a plurality of polar decoded symbols that comprises aplurality of convolutionally encoded symbols (n₁), a first plurality ofinformation symbols (k−n₁), and a plurality of frozen symbols (n−k),wherein k and n₁ are non-negative integers.

The method 1200 includes, at operation 1230, generating a secondplurality of information symbols (k₁) by performing a convolutionaldecoding operation on the plurality of convolutionally encoded symbols,wherein k₁ is a non-negative integer, wherein k₁/n₁ is a rate of theconvolutional encoding operation, and wherein (k₁+k−n₁)/n is a rate ofthe concatenation of the convolutional encoding operation and the polarencoding operation.

The method 1200 includes, at operation 1240, performing a bidirectionalcommunication between the convolutional decoding operation and thesuccessive-cancellation decoding operation, wherein the bidirectionalcommunication comprises decoding information.

In some embodiments, performing the convolutional decoding operation isbased on decoding over a convolutional trellis or tree associated withthe convolutional encoding operation, a Viterbi algorithm, aforward-backward algorithm (FBA) or a BCJR algorithm.

In some embodiments, an operation of the Viterbi algorithm is based onsoft information or symbol likelihoods associated with the noisycodeword vector.

In some embodiments, the successive-cancellation decoding operationcomprises a list decoding operation.

In some embodiments, the noisy codeword vector comprises symbols thatcorrespond to the first plurality of information symbols, the secondplurality of information symbols and a plurality of frozen symbols.

In some embodiments, at least one of the plurality of frozen symbols hasa predetermined value or is based on one or more information symbols inthe noisy codeword vector with indexes less than an index of the atleast one of the plurality of frozen symbols.

In some embodiments, the one or more metrics comprise a location of atleast one of the second plurality of information symbols, and the method1200 further includes the operation of restarting thesuccessive-cancellation decoding operation at the location.

In some embodiments, n₁ is equal to k or k is equal to n.

In some embodiments, the concatenation of the convolutional encodingoperation and the polar encoding operation comprises partitioninginformation bits into the first plurality of information symbols and thesecond plurality of information symbols, performing the convolutionalencoding operation on the first plurality of information symbols togenerate a plurality of encoded symbols, multiplexing the plurality ofencoded symbols, the second plurality of information symbols and theplurality of frozen symbols to generate a plurality of multiplexedsymbols, and performing the polar encoding operation on the plurality ofmultiplexed symbols to generate the codeword.

In some embodiments, the method 1200 further includes the operation ofadjusting the rate k/n based on configuring a number of the plurality offrozen symbols or puncturing the plurality of encoded symbols prior tothe multiplexing.

In some embodiments, the polar encoding operation is based on arecursive “butterfly” computational structure or a multiplication of theplurality of multiplexed symbols by a generator matrix.

In some embodiments, the generator matrix (G) is defined as

${G = {B\begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}}^{\otimes m}},$wherein ⊗ denotes a Kronecker product, B is an n×n bit-reversalpermutation matrix, n=2^(m) is a length of the polar code, and m and nare integers.

In some embodiments, rows of the generator matrix used in the polarencoding operation are selected based on a Bhattacharya parameter or aHamming weight of the rows of the generator matrix.

In some embodiments, the decoding information comprises controlinformation or one or more metrics.

In some embodiments, the one or more metrics comprise one or moremetrics derived from the convolutional decoding operation or one or moremetrics derived from the successive-cancellation decoding operation.

In some embodiments, the control information comprises one or morevalidated estimated coded symbols or one or more invalidated codedsymbols.

In some embodiments, the polar encoding operation comprises a transform,and wherein positions of the plurality of frozen symbols are based onone or more characteristics of the transform.

FIG. 13 illustrates a flowchart of an example method for decoding polarcodes using local feedback. The method 1300 includes, at step 1310,receiving a plurality of information symbols (k), wherein k is apositive integer.

The method 1300 includes, at step 1320, generating a plurality ofconvolutionally encoded symbols (n) by performing a convolutionalencoding operation on the plurality of information symbols and aplurality of frozen symbols (n−k), wherein n is a positive integer.

The method 1300 includes, at step 1330, generating a plurality of polarencoded symbols by performing a polar encoding operation on theplurality of convolutionally encoded symbols, wherein the polar encodingoperation is based on a transform, and wherein a rate of the polarencoding operation is one.

The method 1300 includes, at step 1340, providing the plurality of polarencoded symbols for transmission or storage.

In some embodiments, the method 1300 further includes the operation ofadjusting the rate k/n based on configuring a number of the plurality offrozen symbols or puncturing the plurality of encoded symbols prior tothe multiplexing.

In some embodiments, the polar encoding operation is based on arecursive “butterfly” computational structure or a multiplication of theplurality of multiplexed symbols by a generator matrix.

In some embodiments, the generator matrix (G) is defined as

${G = {B\begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}}^{\otimes m}},$wherein ⊗ denotes a Kronecker product, B is an n×n bit-reversalpermutation matrix, n=2^(m) is a length of the polar code, and m and nare integers.

In some embodiments, rows of the generator matrix used in the polarencoding operation are selected based on a Bhattacharya parameter or aHamming weight of the rows of the generator matrix.

In some embodiments, the decoding information comprises controlinformation or one or more metrics.

In some embodiments, the one or more metrics comprise one or moremetrics derived from the convolutional decoding operation or one or moremetrics derived from the successive-cancellation decoding operation.

In some embodiments, the control information comprises one or morevalidated estimated coded symbols or one or more invalidated codedsymbols.

In some embodiments, the polar encoding operation comprises a transform,and wherein positions of the plurality of frozen symbols are based onone or more characteristics of the transform.

FIG. 14 illustrates a flowchart of another example method for decodingpolar codes using local feedback. The method 1400 includes, at step1410, receiving a noisy codeword vector of length n, the codeword havingbeen generated based on a concatenation of a convolutional encodingoperation and a polar encoding operation and provided to a communicationchannel or a storage channel prior to reception by the decoder, whereinn is a positive integer.

The method 1400 includes, at step 1420, performing asuccessive-cancellation decoding operation on the noisy codeword vectorto generate a plurality of polar decoded symbols (n).

The method 1400 includes, at step 1430, generating a plurality ofinformation symbols (k) by performing a convolutional decoding operationon the plurality of polar decoded symbols, wherein k is a positiveinteger, and wherein k/n is a rate of the concatenation of theconvolutional encoding operation and the polar encoding operation.

The method 1400 includes, at step 1440, performing a bidirectionalcommunication between the successive-cancellation decoding operation andthe convolutional decoding operation.

In some embodiments, the performing the convolutional decoding operationis based on decoding over a convolutional trellis or tree associatedwith the convolutional encoding operation, a Viterbi algorithm, aforward-backward algorithm (FBA) or a BCJR algorithm.

In some embodiments, an operation of the Viterbi algorithm is based onsoft information or symbol likelihoods associated with the noisycodeword vector.

In some embodiments, the bidirectional communication comprises controlinformation or one or more metrics. In an example, the one or moremetrics comprise one or more metrics derived from the convolutionaldecoding operation or one or more metrics derived from thesuccessive-cancellation decoding operation. In another example, thecontrol information comprises one or more validated estimated codedsymbols or one or more invalidated coded symbols.

FIG. 15 is a block diagram representation of a portion of an apparatus,in accordance with some embodiments of the presently disclosedtechnology. An apparatus 1505, such as a base station or a wirelessdevice (or UE), can include processor electronics 1510 such as amicroprocessor that implements one or more of the techniques (including,but not limited to, methods 1200 and 1300) presented in this document.The apparatus 1505 can include transceiver electronics 1515 to sendand/or receive wireless signals over one or more communicationinterfaces such as antenna(s) 1520. The apparatus 1505 can include othercommunication interfaces for transmitting and receiving data. Apparatus1505 can include one or more memories (not explicitly shown) configuredto store information such as data and/or instructions. In someimplementations, the processor electronics 1510 can include at least aportion of the transceiver electronics 1515. In some embodiments, atleast some of the disclosed techniques, modules or functions areimplemented using the apparatus 1505. In some embodiments, the disclosedtechnology can be implemented as part of a computer system (e.g., apersonal computer, a smartphone, a tablet or the like) to enableimproved access to data stored on a memory device associated with thecomputer system. The disclosed embodiments provide enhanced errorcorrection capabilities, which improve the reliability of data storedand accessed in the computer system; at the same time, theseenhancements are achieved by reduced computational complexity of theoperations compared to existing techniques.

Implementations of the subject matter and the functional operationsdescribed in this patent document can be implemented in various systems,digital electronic circuitry, or in computer software, firmware, orhardware, including the structures disclosed in this specification andtheir structural equivalents, or in combinations of one or more of them.Implementations of the subject matter described in this specificationcan be implemented as one or more computer program products, i.e., oneor more modules of computer program instructions encoded on a tangibleand non-transitory computer readable medium for execution by, or tocontrol the operation of, data processing apparatus. The computerreadable medium can be a machine-readable storage device, amachine-readable storage substrate, a memory device, a composition ofmatter effecting a machine-readable propagated signal, or a combinationof one or more of them. The term “data processing unit” or “dataprocessing apparatus” encompasses all apparatus, devices, and machinesfor processing data, including by way of example a programmableprocessor, a computer, or multiple processors or computers. Theapparatus can include, in addition to hardware, code that creates anexecution environment for the computer program in question, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a stand-alone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Computer readable media suitable for storingcomputer program instructions and data include all forms of nonvolatilememory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices. The processor and the memory can be supplemented by, orincorporated in, special purpose logic circuitry.

While this patent document contains many specifics, these should not beconstrued as limitations on the scope of any invention or of what may beclaimed, but rather as descriptions of features that may be specific toparticular embodiments of particular inventions. Certain features thatare described in this patent document in the context of separateembodiments can also be implemented in combination in a singleembodiment. Conversely, various features that are described in thecontext of a single embodiment can also be implemented in multipleembodiments separately or in any suitable subcombination. Moreover,although features may be described above as acting in certaincombinations and even initially claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. Moreover, the separation of various system components in theembodiments described in this patent document should not be understoodas requiring such separation in all embodiments.

Only a few implementations and examples are described and otherimplementations, enhancements and variations can be made based on whatis described and illustrated in this patent document.

What is claimed is:
 1. A system for improving an error correctioncapability of a decoder comprising a successive-cancellation decoder anda convolutional decoder, the system comprising: a processor; and anon-transitory memory including instructions stored thereon, wherein theinstructions upon execution by the processor cause the processor to:receive a noisy codeword vector of length n, the codeword having beengenerated based on a concatenation of a convolutional encoding operationand a polar encoding operation and provided to a communication channelor a storage channel prior to reception by the decoder, wherein n is apositive integer; perform, using the successive-cancellation decoder, asuccessive-cancellation decoding operation on the noisy codeword vectorto generate a plurality of polar decoded symbols that comprises aplurality of convolutionally encoded symbols (n₁), a first plurality ofinformation symbols (k−n₁), and a plurality of frozen symbols (n−k),wherein k and n₁ are non-negative integers; generate a second pluralityof information symbols (k₁) by performing, using the convolutionaldecoder, a convolutional decoding operation on the plurality ofconvolutionally encoded symbols, wherein k₁ is a non-negative integer,wherein k₁/n₁ is a rate of the convolutional encoding operation, andwherein (k₁+k−n₁)/n is a rate of the concatenation of the convolutionalencoding operation and the polar encoding operation; and perform abidirectional communication between the convolutional decoder and thesuccessive-cancellation decoder, wherein the bidirectional communicationcomprises decoding information.
 2. The system of claim 1, wherein theperforming the convolutional decoding operation is based on decodingover a convolutional trellis or tree associated with the convolutionalencoding operation, a Viterbi algorithm, a forward-backward algorithm(FBA) or a BCJR algorithm.
 3. The system of claim 2, wherein anoperation of the Viterbi algorithm is based on soft information orsymbol likelihoods associated with the noisy codeword vector.
 4. Thesystem of claim 1, wherein the successive-cancellation decodingoperation comprises a list decoding operation.
 5. The system of claim 1,wherein the noisy codeword vector comprises symbols that correspond tothe first plurality of information symbols, the second plurality ofinformation symbols and a plurality of frozen symbols.
 6. The system ofclaim 5, wherein at least one of the plurality of frozen symbols has apredetermined value or is based on one or more information symbols inthe noisy codeword vector with indexes less than an index of the atleast one of the plurality of frozen symbols.
 7. The system of claim 1,wherein the one or more metrics comprise a location of at least one ofthe second plurality of information symbols, and wherein theinstructions upon execution by the processor further cause the processorto: restart the successive-cancellation decoding operation at thelocation.
 8. The system of claim 1, wherein n₁ is equal to k.
 9. Thesystem of claim 1, wherein k is equal to n.
 10. The system of claim 1,wherein the instructions upon execution by the processor further causethe processor, as part of performing the concatenation of theconvolutional encoding operation and the polar encoding operation, to:partition information bits into the first plurality of informationsymbols and the second plurality of information symbols; perform theconvolutional encoding operation on the first plurality of informationsymbols to generate a plurality of encoded symbols; multiplex theplurality of encoded symbols, the second plurality of informationsymbols and the plurality of frozen symbols to generate a plurality ofmultiplexed symbols; and perform the polar encoding operation on theplurality of multiplexed symbols to generate the codeword.
 11. Thesystem of claim 10, wherein the instructions upon execution by theprocessor further cause the processor to: adjust the rate k/n based onconfiguring a number of the plurality of frozen symbols or puncturingthe plurality of encoded symbols prior to the multiplexing.
 12. Thesystem of claim 10, wherein the polar encoding operation is based on arecursive “butterfly” computational structure or a multiplication of theplurality of multiplexed symbols by a generator matrix.
 13. The systemof claim 12, wherein the generator matrix (G) is defined as$G = {B\begin{bmatrix}1 & 0 \\1 & 1\end{bmatrix}}^{\otimes m}$ wherein ⊗ denotes a Kronecker product,wherein B is an n×n bit-reversal permutation matrix, wherein n=2^(m) isa length of the polar code, and wherein m and n are integers.
 14. Thesystem of claim 13, wherein rows of the generator matrix used in thepolar encoding operation are selected based on a Bhattacharya parameteror a Hamming weight of the rows of the generator matrix.
 15. The systemof claim 1, wherein the decoding information comprises controlinformation or one or more metrics.
 16. The system of claim 13, whereinthe one or more metrics comprise one or more metrics derived from theconvolutional decoding operation or one or more metrics derived from thesuccessive-cancellation decoding operation.
 17. The system of claim 13,wherein the control information comprises one or more validatedestimated coded symbols or one or more invalidated coded symbols. 18.The system of claim 10, wherein the polar encoding operation comprises atransform, and wherein positions of the plurality of frozen symbols arebased on one or more characteristics of the transform.
 19. Anon-transitory computer-readable storage medium having instructionsstored thereupon for improving an error correction capability of adecoder, comprising: instructions for receiving a noisy codeword vectorof length n, the codeword having been generated based on a concatenationof a convolutional encoding operation and a polar encoding operation andprovided to a communication channel or a storage channel prior toreception by the decoder, wherein n is a positive integer; instructionsfor performing a successive-cancellation decoding operation on the noisycodeword vector to generate a plurality of polar decoded symbols thatcomprises a plurality of convolutionally encoded symbols (n₁), a firstplurality of information symbols (k−n₁), and a plurality of frozensymbols (n−k), wherein k and n₁ are non-negative integers; instructionsfor generating a second plurality of information symbols (k₁) byperforming a convolutional decoding operation on the plurality ofconvolutionally encoded symbols, wherein k₁ is a non-negative integer,wherein k₁/n₁ is a rate of the convolutional encoding operation, andwherein (k₁+k−n₁)/n is a rate of the concatenation of the convolutionalencoding operation and the polar encoding operation; and instructionsfor performing a bidirectional communication between the convolutionaldecoding operation and the successive-cancellation decoding operation,wherein the bidirectional communication comprises decoding information.20. The storage medium of claim 19, wherein the performing theconvolutional decoding operation is based on decoding over aconvolutional trellis or tree associated with the convolutional encodingoperation, a Viterbi algorithm, a forward-backward algorithm (FBA) or aBCJR algorithm.
 21. The storage medium of claim 20, wherein an operationof the Viterbi algorithm is based on soft information or symbollikelihoods associated with the noisy codeword vector.
 22. The storagemedium of claim 19, wherein the successive-cancellation decodingoperation comprises a list decoding operation.
 23. The storage medium ofclaim 19, wherein either n₁ is equal to k or k is equal to n.